Ask Dr. Math

High School Archive

Browse High School Basic Algebra
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    Solving simple linear equations.
    Positive/negative integer rules.
    Mixture problems.
    Quadratic equations.
    Absolute value.
    Completing the square.
    Direct and indirect variation.
    Inequalities and negative numbers.

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Triangle Perimeter [07/20/1997]

How many triangles have sides whose lengths total 15 units?

A Trick for Solving Equations with Fractions [11/12/1995]

Help solving an equation with fractions.

Trinomial Expansion [12/08/1998]

I need to expand (1+x+x^2)^n to four terms. Is there a formula?

Triples of Numbers [7/23/1996]

Find two triples of positive integers such that their sum equals their product.

Two Airplanes [08/28/1997]

Draw a diagram using coordinate system with origin O - how far apart are the airplanes? Express the answer in radical form.

Two Algebra Word Problems [3/23/1996]

Two people decide to open a music store. A building is available for rent for $6000 per year. Heat, electricity, taxes, etc... are estimated at $5000 per year. Each person wants to make $7 per hour...

Two Cars [08/11/1999]

How long does it take car B to draw even with car A, and how far does car B travel?

Two Equations, Two Unknowns [7/27/1996]

How do I solve for x and y in terms of a and b for the equations x^4 + y^4 = b^4 and x + y = a?

Two Hot-Air Balloons [10/21/1996]

The height h of the first balloon at time t is h(t)=50+34t. The height g of the second balloon is g(t)=400-16t. When will they be at the same height and how high will they be?

Two New Variables [10/25/2002]

I am trying to calculate using the Owens-Wendt equation. The equation will give two equations with two unknowns. I am unable to solve them, because of the x^1/2 and y^1/2 powers.

Two Numbers with Equal Sum, Product, and Quotient? [05/07/2007]

Find two numbers such that when you find the sum, product, and quotient of the two numbers they are all equal.

Two Values of X [03/20/2002]

Two numbers differ by 7 and the sum of their squares is 389. Find the two numbers.

Understanding Parallel and Perpendicular [03/23/1998]

Can you explain parallel and perpendicular lines and slopes?

Understanding Rectangle Area and Perimeter [11/08/2002]

True or false: if the perimeter of a rectangle increases, the rectangle's area always also increases.

Unraveling an Inverse Function [11/30/2001]

I got this question wrong: y = (-5x - 2) / (-x + 1).

Unusual Simultaneous Equations [02/06/2003]

A = a*(b+c)/(a+b+c); B = b*(a+c)/(a+b+c); C = c*(a+b)/(a+b+c). A, B, and C are known. Solve for a, b, and c.

Usefulness of General Form of Quadratic Equation [03/24/2005]

When is it useful to know the general form of a quadratic? Why wouldn't I rather know the vertex form or factored form since those tell me something about the graph?

Using Cardan's Formula to find Real Roots [04/13/2000]

How can I show that, although the cubic equation x^3 - 6x = 4 has three real solutions, Cardan's formula can find them by subtracting appropriate cube roots of complex numbers?

Using Euclid's Algorithm with Three Numbers [11/05/2003]

How do I find the GCD of three integers using Euclid's Algorithm? I am not sure where you plug the third integer into the algorithm.

Using Graphs to Solve Problems [12/29/1996]

How do you use graphs to find the answers to questions?

Using Interval Notation to Express Answers [12/03/1996]

Solve for x in the inequality x/(x+3) >= 1/(x-1) and express the answer in interval notation.

Using Relative Primes [12/07/1996]

Given a floor 105 tiles wide and 135 tiles long, how many tiles will a diagonal drawn from one corner to the opposite corner intersect?

Using Substitution [10/05/1998]

Can you explain the use of substitution in your cubic polynomial frequently asked question?

Using the Discriminant [07/19/2000]

How can I determine the character of the solution of the equation x^2+2x+6=0 in the complex numbers without solving it?

Using the Distance Formula [05/03/1997]

Given a line going through the points (10,10) and (20,15), find the coordinates of a third point on the line that is 3 units from the point (10,10).

Using the Slope-Intercept Formula [9/27/1995]

How can I write an equation for the line when I have the x-intercept and the slope?

Using Trial Solutions [07/28/1999]

How can I solve the system of equations x-sqrt(yz)=42, y-sqrt(xz)=6, z- sqrt(xy)=-30 to find x, y and z?

Value Excluded from the Domain [5/23/1996]

f(x) = 3x^3 + x^2 - 2x/(9x^2 - 4).

The Value of ABS[?] [08/24/2010]

Absolute value symbols keep appearing out of nowhere. Doctor Ian demystifies.

A Variable and Its Reciprocal, Each Cubed [11/20/2010]

Given a variable plus its multiplicative inverse, a student wonders how to evaluate a sum of the same form, but where each of the unknowns has been cubed. Doctor Ali suggests starting with a simpler problem before going on to show a few steps of algebraic expansion and re-arrangement.

Velocity and Time [02/11/1999]

A car averages 30 km/hr the first half of a trip, and 50 km/hr the second half of the trip. What is the car's average velocity for the trip?

A Velocity-Time Problem [02/16/1999]

If you drive 40 mph instead of 30 mph, you save 30 seconds per mile, but if you make the same 10 mph increase from 70 mph to 80 mph, you only save 5.6 seconds per mile. Why?

Vertex of a Parabola [04/16/2002]

Finding the y-coordinate of the vertex of a parabola.

Vertical and Horizontal Asymptotes [6/3/1996]

What are the vertical and horizontal asymptotes?

Visualizing 1 + 1/x [10/10/2003]

Show that the sum of a positive number and its reciprocal is at least 2.

Walking the Shortest Distance [10/07/2002]

A single boy lives in each of n equally spaced houses on a straight line. At what point should the boys meet so that the sum of the distances that they walk from their houses is as small as possible?

Water Level in a Well [10/09/2001]

A man wishes to determine the water level in a deep well. He drops a stone into the well and hears it hit the water 3 seconds later...

Water Main Section Lengths [01/09/2003]

A water main is being laid using pipe that comes in either 18-foot or 20-foot sections. The water main will require 14 fewer sections of 20- foot pipe than if 18-foot sections are used. Find the total length of the water main.

Weighted Average of the Velocities [01/30/2003]

Finding the average velocity for each of the three phases of the trip...

What are Piecewise Functions? [05/13/2001]

I know piecewise functions are based on expressions between specific intervals, but I do not know how to describe this function family.

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